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Kronecker :: decomposeModule

decomposeModule -- decompose a module into a direct sum of simple modules

Synopsis

Description

This function decomposes a module into a direct sum of simple modules, given some fairly strong assumptions on the ring which acts on the ring which acts on the module. This ring must only have two variables, and the square of each of those variables must kill the module.
i1 : Q = ZZ/101[x,y]

o1 = Q

o1 : PolynomialRing
i2 : R = Q/(x^2,y^2)

o2 = R

o2 : QuotientRing
i3 : M = coker random(R^5, R^8 ** R^{-1})

o3 = cokernel | -29x    -47x+32y 21x+17y 19x+18y  -29x-32y 41x-45y -27x-40y -32x-33y |
              | 3x-45y  24x+39y  -4x+43y -4x+18y  -35x+8y  19x+44y -34x-34y -34x-34y |
              | -13x-8y 30x+4y   29x-26y -39x-19y 37x-26y  34x-20y 13x-31y  -32x+7y  |
              | 9x+13y  -31x-4y  -3x+4y  4x-38y   -6x-9y   27x-35y -45x+19y -44x-29y |
              | -14x-7y 39x-6y   -46x+8y 49x-15y  -11x-20y 29x-6y  16x+10y  33x-27y  |

                            5
o3 : R-module, quotient of R
i4 : (N,f) = decomposeModule M

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -17 21  0   38  -12 |)
               | 0 0 x 0 y 0 0 0 |  | -41 -29 -43 -45 31  |
               | 0 0 0 y x 0 0 0 |  | -12 -38 9   39  -29 |
               | 0 0 0 0 0 x 0 y |  | 45  -1  -44 6   -42 |
               | 0 0 0 0 0 0 y x |  | 1   0   0   0   0   |

o4 : Sequence
i5 : components N

o5 = {cokernel | y x |, cokernel | x 0 y |, cokernel | x 0 y |}
                                 | 0 y x |           | 0 y x |

o5 : List
i6 : ker f == 0

o6 = true
i7 : coker f == 0

o7 = true

Ways to use decomposeModule :